Almost K\"ahler metrics and pp-wave spacetimes

TL;DR

This paper establishes a deep connection between almost K"ahler metrics and Lorentzian pp-wave spacetimes, constructing new examples and showing that every Lorentzian metric locally admits an almost K"ahler limit via Penrose's plane wave limit.

Contribution

It constructs families of complete almost K"ahler metrics from pp-waves and demonstrates that every Lorentzian metric admits a local almost K"ahler limit using Penrose's construction.

Findings

Constructed new almost K"ahler metrics from pp-waves.

Showed all Lorentzian metrics have local almost K"ahler limits.

Provided examples with constant negative scalar curvature.

Abstract

We establish a one-to-one correspondence between a class of strictly almost K\"ahler metrics on the one hand, and Lorentzian pp-wave spacetimes on the other; the latter metrics are well known in general relativity, where they model radiation propagating at the speed of light. Specifically, we construct families of complete almost K\"ahler metrics by deforming pp-waves via their propagation wave vector. The almost K\"ahler metrics we obtain exist in all dimensions $2n \geq 4$, and are defined on both $\mathbb{R}^{2n}$ and $\mathbb{S}^1\times\mathbb{S}^1 \times M$, where $M$ is any closed almost K\"ahler manifold; they are not warped products, they include noncompact examples with constant negative scalar curvature, and all of them have the property that their fundamental 2-forms are also co-closed with respect to the Lorentzian pp-wave metric. Finally, we further deepen this relationship between almost K\"ahler and Lorentzian geometry by utilizing Penrose's "plane wave limit," by which every spacetime has, locally, a pp-wave metric as a limit: using Penrose's construction, we show that in all dimensions $2n \geq 4$, every Lorentzian metric admits, locally, an almost K\"ahler metric of this form as a limit.